Coupled MIMO finite-Hankel model reduction
==========================================

.. admonition:: Tutorial goal

   Reduce a genuinely coupled MIMO state-space system with the finite block-Hankel baseline.

.. note::

   New to the terminology? See the :doc:`lattice DSP concept map <../../algorithms/concept_map>` and the :doc:`causality/data-use guide <../../theory/causality_and_data_use>` for how online, offline, block, and MIMO examples should be read.

Context
-------

The diagonal MIMO tutorial shows that independent SISO filters are a special case.  This
tutorial uses a dense, stable state-space system where each input can affect each output.
The goal is to validate the practical MIMO finite-section baseline on a coupled
system while keeping it separate from matrix AAK/Nehari algorithms.

The reducer works with Markov matrices and returns a reduced state-space realization.  This
is the natural representation for MIMO; scalar numerator/denominator coefficients are not
forced onto a multivariable system.

Key idea and equations
----------------------

A coupled MIMO state-space model has

.. math::

   x_{n+1}=Ax_n+Bu_n,\qquad y_n=Cx_n+Du_n.

Its Markov matrices are

.. math::

   M_0=D,\qquad M_k=CA^{k-1}B\quad(k\ge 1).

The finite block-Hankel reducer constructs a reduced model
``(A_r,B_r,C_r,D)`` from the leading singular directions of the block-Hankel matrix.

How to read the result
----------------------

Look for nonzero off-diagonal channels, block-Hankel singular-value decay, decreasing Markov/output error with order, and reduced state radii below one.

Run command
-----------

.. code-block:: bash

   python examples/mimo_coupled_model_reduction.py

Source code
-----------

.. literalinclude:: ../../../examples/mimo_coupled_model_reduction.py
   :language: python
   :linenos: